On special unipotent orbits and Fourier coefficients for automorphic forms on symplectic groups

Fourier coefficients of automorphic representations π of Sp2n(A) are attached to unipotent adjoint orbits in Sp2n(F), where F is a number field and A is the ring of adeles of F. We prove that for a given π, all maximal unipotent orbits that give nonzero Fourier coefficients of π are special, and pro...

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Bibliographic Details
Published inJournal of number theory Vol. 146; pp. 343 - 389
Main Authors Jiang, Dihua, Liu, Baiying
Format Journal Article
LanguageEnglish
Published Elsevier Inc 01.01.2015
Subjects
Automorphic forms
Fourier coefficients
Unipotent orbits
secondary
Unipotent orbits
Automorphic forms
Fourier coefficients
primary
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Summary:Fourier coefficients of automorphic representations π of Sp2n(A) are attached to unipotent adjoint orbits in Sp2n(F), where F is a number field and A is the ring of adeles of F. We prove that for a given π, all maximal unipotent orbits that give nonzero Fourier coefficients of π are special, and prove, under a well-acceptable assumption, that if π is cuspidal, then the stabilizer attached to each of those maximal unipotent orbits is F-anisotropic as algebraic group over F. These results strengthen, refine and extend the earlier work of Ginzburg, Rallis and Soudry on the subject. As a consequence, we obtain constraints on those maximal unipotent orbits if F is totally imaginary, further applications of which to the discrete spectrum with the Arthur classification will be considered in our future work.
ISSN:0022-314X
1096-1658
DOI:10.1016/j.jnt.2014.03.002